On 12/17/2012 11:47 AM, Bruno Marchal wrote:


On 16 Dec 2012, at 20:28, meekerdb wrote:

On 12/16/2012 2:31 AM, Bruno Marchal wrote:
No. With the CTM the ultimate truth is arithmetical truth, and we cannot really define it (with the CTM). We can approximate it in less obvious ontologies, like second order logic, set theory, etc. But with CTM this does not really define it. Don't confuse truth, and the words pointing to it. Truth is always beyond words, even the ultimate 3p truth.

What would it mean to 'define truth'? We can define 'true' as a property of sentence that indicates a fact.

That's the best definition of some useful local truth. But when doing metaphysics, you have to replace facts by "facts in some model/reality".

OK. But then it's "True relative to the model." and it's not necessarily The 
Truth.



But I'm not sure how to conceive of defining mathematical 'true'.

It is the object of model theory. You always need to add more axiom in a theory to handle its model. You cannot define the notion of truth-about-set in ZF, but you can define truth-about-set in ZF in the theory ZF +kappa (existence of inaccessible cardinals).

PA can define all the notion of truth for the formula with a bounded restriction of the quantification.


So what is that definition?





Does it just mean consistent with a set of axioms,

No. That means only having a model. true in some reality. But for arithmetic "true" means satisfied by the usual structure (N, +, *).



i.e. not provably false?

How is not provably false different from 'satisfied by the usual structure'? Can you give an example?


That just consistent.

I would think it was incompleteness. Consistency means not being able to prove every proposition. But in a consistent system there can be propositions that are neither provable nor disprovable. Are those true?

Brent

True entails consistency, but consistency does not entail truth.

Bruno

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